Lesson Video: Convex Mirrors Science

Video Transcript

In this video, we will learn how to describe the reflection of light rays from a convex mirror. First, let’s see how a convex mirror looks.

Seen from the side, such a mirror could look this way. Were we to shine a ray of light on the mirror, it would reflect or bounce off. Let’s imagine that instead of looking at this mirror from the side, we look at it from above like this. From that perspective, here’s how the mirror would appear. This mirror is called convex because the middle of the mirror, here, is closer to an observer than the edges of the mirror are. We could say that the middle of the mirror bends toward the observer. This is a convex mirror.

At the same time, this is a spherical mirror. We can say that because the surface of the mirror, here, is a small part of a sphere. The mirror’s full name is a convex spherical mirror. This point here is the center of the sphere that our mirror helps to make. It has a special name, the center of curvature. The distance between the center of curvature in any point on the surface of the mirror is the same. That is, this distance here equals this distance here, equals this distance here, equals this distance here, and so on. The name for the distance between the center of curvature and a point on the mirror surface is the radius of curvature.

Now, we know that the job of a mirror is to reflect light. Say we have some incoming light rays that are parallel to one another. When these rays reach the mirror, they’ll bounce off. This top ray of light reaches the mirror at the center of the mirror surface. What will happen is that this ray reflects directly backward along the path that it traveled to reach the mirror. The next parallel ray of light, once it reaches the mirror, will reflect off its surface like this. And then the last ray will bounce off the mirror in this direction. The three reflected rays never meet. In fact, they get farther and farther apart as they travel. Because the reflected rays don’t meet, they don’t form an image.

So then, what is the image we see when we look at an object’s reflection in a mirror? Say that we take our three reflected rays, and we trace them backward. These dashed lines do meet at a point. Since this is where the reflected rays would meet if they could travel back into the mirror, we call this the focal point. Whenever we look at a mirror and see an image, that image seems to be at the mirror’s focal point.

So far, we’ve identified three important points on this diagram: the center of curvature, the focal point, and this point here, which is the center of the mirror surface. The distance between the center of the surface of the mirror and the focal point is called the focal length. For any convex spherical mirror, the focal length is exactly one-half the radius of curvature. Knowing all this about convex spherical mirrors, let’s look now at a few examples.

Below is a ray diagram for a convex mirror. Which one of the five locations along the optical access represents the focal point of the spherical mirror?

Since this mirror is convex, that means an observer would be on this side of the mirror. Since the mirror is spherical, that means its surface is part of a larger sphere. In this example, we have five locations, one, two, three, four, five, along what is called the optical axis. That axis follows this line that goes right through the center of the mirror. We want to figure out which of these five locations represents the focal point of the mirror. Notice that the diagram shows these three parallel rays of light running into the mirror. The top ray reflects off in this direction, the middle ray reflects straight back along the path it came, and the bottom ray reflects off the mirror like this.

To find the location of the focal point, we’ll trace these three reflected rays backward. The top ray is traced backward like this, the middle ray like this, and the bottom ray this way. We’ve only traced back far enough to find the point where the rays meet, where they cross. That point is here, and this is the focal point of the mirror. We see this is at the location marked four. Location four is the mirror’s focal point.

Let’s look now at another example.

The radius of curvature of a convex mirror is five centimeters. Which one of the following sentences about the focal length is correct? (A) The focal length is five centimeters and is the distance from the center of the surface of the mirror to the focal point. (B) The focal length is five centimeters and is the distance from the center of the surface of the mirror to the center of curvature. (C) The focal length is 2.5 centimeters and is the distance from the center of the surface of the mirror to the center of curvature. (D) The focal length is 2.5 centimeters and is the distance from the center of the surface of the mirror to the focal point.

Knowing that we have a convex mirror with a radius of curvature of five centimeters, we can clear space at the top of our screen and sketch this mirror. Let’s say that this is our convex mirror, and here is the mirror center of curvature. The distance between this point and the point at the center of the mirror here is called the radius of curvature. In this example, the radius of curvature is given as five centimeters. Now, all of our answer options describe the mirror’s focal length. This is different from the radius of curvature.

If we sketch it on our diagram, this mirror’s focal point would be a point here. It’s the point halfway between the center of curvature and the center of the surface of the mirror. Then, the distance between the center of the surface of the mirror and our focal point, that distance is called the focal length of the mirror. As an equation, we can say that focal length equals one-half times the radius of curvature. Since the radius of curvature of this mirror is five centimeters, then the focal length must be half of that, or 2.5 centimeters.

Only two of our answer options, (C) and (D), have the focal length at 2.5 centimeters. This means we can cross out options (A) and (B). The difference between our two remaining answer choices is that one says the focal length is the distance from the center of the surface of the mirror to the center of curvature, while the other has it as the distance from the center of the surface of the mirror to the focal point. To see the difference between these choices, let’s look at an up close view of our sketch.

Okay, in this view, we have our center of curvature, the focal point of the mirror, and then here is the center of the surface of the mirror. Answer option (C) says that the focal length of this mirror is measured as this distance. We can see, though, that this can’t be correct. That’s because this distance in pink is the radius of curvature, five centimeters. The focal length is one-half that distance. The true focal length is the distance from the center of the surface of the mirror to the focal point. This is described by option (D). The focal length is 2.5 centimeters and is the distance from the center of the surface of the mirror to the focal point.

Let’s look now at one last example.

Which one of the following sentences is the correct description for what happens to parallel rays incident on a convex mirror? (A) They will continue undisturbed. (B) They will be focused at a point which is called the focal point. (C) They will not be focused at a point, but the mirror will still have a focal point. (D) They will not be focused at a point, and the mirror will have no focal point.

To see which answer is correct, let’s clear some space at the top of our screen. And we can draw parallel rays incident on a convex mirror. Because these rays are reaching a mirror, they will reflect off of it. The center ray will bounce straight backward in this direction, while the top and bottom rays will reflect like this.

Looking again at our answer choices, we see that answer (A) can’t be correct. These rays do not continue undisturbed; instead, they reflect off the mirror. Option (B) says that the rays are focused to a point, but we see that these reflected rays travel farther and farther away from one another over time. Choice (B) can’t be correct either. To figure out which of options (C) and (D) is correct, we need to know whether this mirror has a focal point. To find out, let’s trace the reflected rays backward using dashed lines.

Tracing the center reflected right back would give this line. And then the top and bottom reflected rays would be traced backward like this. These rays do cross, which means there is a focal point. This means choice (C) is our answer. The reflected rays will not be focused at a point, but the mirror will still have a focal point. It’s the point we’ve identified in orange.

Let’s now finish this lesson by reviewing a few key points. In this video, we learned about convex spherical mirrors. These are convex because the center of the mirror is closer to an observer than the edges. And they are spherical because the surface of the mirror is part of the surface of a sphere. The center of that sphere is called the center of curvature, and the distance between the center of curvature and the center of the surface of the mirror is called the radius of curvature. When parallel rays of light are incident on the mirror, those rays reflect back and do not cross. But if we trace the reflected rays backward, those trace lines do intersect; they meet at the focal point of the mirror. The distance between the focal point and the center of the surface of the mirror is called the focal length. The focal length is equal to one-half the radius of curvature of the mirror. This is a summary of convex mirrors.